: 1.  1
:    ∫((1-x^7)^(1/3)-(1-x^3)^(1/7))dx
:     0

 1                    1 (1-x^7)^(1/3)   1 (1-y^3)^(1/7)   1
∫(1-x^7)^(1/3) dx = ∫∫ dydx       = ∫∫ dxdy       = ∫(1-y^3)^(1/7) dy
 0                    0 0               0 0               0

所以說 原式 = 0

: 2. lim{∫([bx+a(1-x)]^n)dx)}^(1/n),b>a>0.
:    n→0

                       [(b-a)x + a]^(n+1)
∫[(b-a)x + a]^n dx = -------------------- + c
                           (n+1)(b-a)

這題是不是有缺條件啊...?

: 3.一曲面x^2/4+y^2+z^2/9=3,求其在(-2,1,3)這點的tangent plane和normal line.

f(x) = x^2/4 + y^2 + z^2/9 - 3, del[f(x)] = (x/2, 2y, 2z/9)

del[f(-2,1,3)] = (-1,2,2/3) = (-3,6,2)

tangent plane: -3(x+2) + 6(y-1) + 2(z-3) = 0, 3x-6y-2z+18=0

              x+2     y-1     z-3
normal line: ----- = ----- = -----
               -3      6       2

: 4.一個碗裡面有水,水蒸發的速率和水面面積(就是水和空氣接觸的面積)成正比,請證明水
:   面下降的速率(水深減少的速率)為定值(和碗的形狀無關).

  -dV         -Adz         -dz
 ----- = kA, ------ = kA, ----- = v = k, k為一常數
   dt          dt           dt

: 5. ∞
:    Σ (1/(1+n^2)) 是否收斂?
:    n=1

          1
       -------
        1+n^2               1
  lim --------- =  lim ----------- = 1, 原函數收斂
 n->∞    1       n->∞   1
        -----           ----- + 1
         n^2             n^2

: 6.一曲線為x^2+y^2=1及x-y+z=1的交線,而f(x,y,z)=x+2y+3z,求在曲線上f的最大值.

del(x+2y+3z) = a[del(x^2+y^2-1)] + b[del(x-y+z-1)]

1 = 2xa + b, 2 = 2ya - b, 3 = b

-1 = xa, 5/2 = ya, x = (-2/5)y

y = +-5/(29)^(1/2), x = -+2/(29)^(1/2), z = 1 +- 7/(29)^(1/2)

f(-2/(29)^(1/2), 5/(29)^(1/2), 1 + 7/(29)^(1/2)) = 3 + (29)^(1/2) = max

: 7.R為y^2=4-4x和y^2=4+4x所圍成的區域,其中y≧0,求∫∫ydA
:                                                  R

           0 (4+4x)^(1/2)   1 (4-4x)^(1/2)
∫∫ydA = ∫ ∫ ydydx    + ∫ ∫ ydydx    = 1 + 1 = 2
  R       -1  0             0  0

--
※ 編輯: hsnuyi (118.168.236.205 臺灣), 09/12/2019 01:07:58